Calculate the apothem of a regular regular triangular pyramid, if the side of the base is a

| May 29, 2021 | education

Calculate the apothem of a regular regular triangular pyramid, if the side of the base is a, the height of the pyramid is h

At the base of the pyramid is a regular triangle AB = BC = AC = a cm.

The height BH in the triangle is also the bisector and median, then CH = AB = a / 2 cm.

Then BH ^ 2 = AB ^ 2 – CH ^ 2 = a ^ 2 – a ^ 2/4 = 3 * a ^ 2/4.

BH = a * √3 / 2 cm.

Since the medians of the triangle at the intersection point are divided in the ratio of 2/1, then BO / OH = 2/1.

BО = 2 * OH.

BО + ОН = BН.

BH = 3 * OH.

OH = BH / 3 = (a * √3 / 2) / 3 = a * √3 / 6 cm.

In a right-angled triangle DOH DH ^ 2 = DO ^ 2 + OH ^ 2 = h ^ 2 + (a * √3 / 6) ^ 2 = h ^ 2 + a ^ 2 * 3/36 =

h ^ 2 + a ^ 2/12.

DH = √ (h ^ 2 + a ^ 2/12) cm.

Answer: Apothem of the pyramid is equal to √ (h2 + a2 / 12) cm.



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